# Ordering on Natural Numbers is Compatible with Addition/Corollary

## Corollary to Ordering on Natural Numbers is Compatible with Addition

Let $a, b, c, d \in \N$ where $\N$ is the set of natural numbers.

Then:

$a > b, c > d \implies a + c > b + d$

## Proof

 $\displaystyle a$ $>$ $\displaystyle b$ $(1):\quad$ $\displaystyle \leadsto \ \$ $\displaystyle a + c$ $>$ $\displaystyle b + c$ Ordering on Natural Numbers is Compatible with Addition

 $\displaystyle c$ $>$ $\displaystyle d$ $(2):\quad$ $\displaystyle \leadsto \ \$ $\displaystyle b + c$ $>$ $\displaystyle b + d$ Ordering on Natural Numbers is Compatible with Addition

Finally:

 $\displaystyle a + c$ $>$ $\displaystyle b + c$ from $(1)$ $\displaystyle b + c$ $>$ $\displaystyle b + d$ from $(2)$ $\displaystyle \leadsto \ \$ $\displaystyle a + c$ $>$ $\displaystyle b + d$ Ordering on Natural Numbers is Trichotomy

$\blacksquare$